Solution to Linear First Order ODE with Constant Coefficients/Proof 1

Theorem

A linear first order ODE with constant coefficients in the form:

$(1): \quad \dfrac {\d y} {\d x} + a y = \map Q x$

has the general solution:

$\ds y = e^{-a x} \paren {\int e^{a x} \map Q x \rd x + C}$

Proof

From the Product Rule for Derivatives:

\(\ds \map {\frac \d {\d x} } {e^{a x} y}\)

\(=\)

\(\ds e^{a x} \cdot \dfrac {\d y} {\d x} + y \cdot a e^{a x}\)

\(\ds \)

\(=\)

\(\ds e^{a x} \paren {\dfrac {\d y} {\d x} + a y}\)

Hence, multiplying $(1)$ all through by $e^{\int a \rd x}$:

$\map {\dfrac \d {\d x} } {e^{a x} y} = e^{a x} \map Q x$

Integrating with respect to $x$ now gives:

$\ds e^{a x} y = \int e^{a x} \map Q x \rd x + C$

whence we get the result by dividing by $e^{a x}$.

$\blacksquare$

Sources

• 1958: G.E.H. Reuter: Elementary Differential Equations & Operators ... (previous) ... (next): Chapter $1$: Linear Differential Equations with Constant Coefficients: $\S 1$. The first order equation: $\S 1.2$ The integrating factor: $(7)$